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CE 191: Civil and Environmental EngineeringSystems Analysis

LEC 03 : Graphical Solutions to LP

Professor Scott MouraCivil & Environmental EngineeringUniversity of California, Berkeley

Fall 2014

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 1

Graphical Solutions of Linear Programs

Example:

min J = 140x1 + 160x2

s. to 2x1 + 4x2 ≤ 28

5x1 + 5x2 ≤ 50

x1 ≤ 8

x2 ≤ 6

x1 ≥ 0

x2 ≥ 0

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 2

Construction of the feasible set

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 3

Construction of the feasible set

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Construction of the feasible set

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Construction of the feasible set

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Construction of the feasible set

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Construction of the feasible set

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Construction of the feasible set

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Feasible Set Final Result

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 10

Isolines

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Isolines

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Isolines

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Gradient of the cost function

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Uniqueness (or not) of the cost function

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Features of the feasible set

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Features of the feasible set

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Features of the feasible set

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Features of the feasible set

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Feasible set is unbounded

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Objective function might be unbounded too

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Objective function might be bounded

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 22

Optimum may be non-unique

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 23

Feasible set might be empty

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Feasible set might be empty

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Feasible set might be empty

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 26

Feasible set might be empty

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 27

Constraint domination

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Constraint domination

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Constraint domination

dominated

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 30

Graphical solution of LPs: A General Method

Write  your  LP  

Successively  eliminate  half-­‐spaces  

Feasible  Set  Empty?  

Problem  Infeasible  

Objec?ve  Bounded?  

Infinite  Solu?on  Solu?on  

Unique?  

Feasible  Set  Bounded?  

YES   NO  

NO  

NO  YES  

Corner  Point  Solu?on  

Boundary  Solu?on  

YES   NO  

YES  

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 31

Insights from Graphical LP

Linear constraints Ax ≤ b form feasible set (possibly empty)

Feasible set is a (possibly unbounded) convex polytope

Optimal solution exists along edges (corner point or line segment)

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 32

Danzig’s Simplex Algorithm

1 Define feasible set

2 Start at vertex. Move along vertices until obj. fcn. stops decreasing

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 33

Example of Simplex Algorithm

Recall the LP problem:

max J = 140x1 + 160x2

s. to 2x1 + 4x2 ≤ 28

5x1 + 5x2 ≤ 50

x1 ≤ 8

x2 ≤ 6

x1 ≥ 0

x2 ≥ 0

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 34

Feasible Set Final Result

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 35

Start at a Vertex

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 36

Jump to adjacent vertex

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Stop when objective stops decreasing

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 38

Additional Reading

Revelle

Chapter 3 - A Graphical Solution Procedure and Further Examples

Simplex Algorithm

Revelle Chapter 4 - The Simplex Algorithm for Solving Linear Programs

Prof. Moura | UC Berkeley CE 191 | LEC 03 - Graphical Solns to LP Slide 39